Mathematical Thinking Opens Doors for Students with Learning Disabilities

We are convinced that not only are students with learning disabilities capable of thinking mathematically, but mathematical thinking serves as a powerful learning support for those who struggle in mathematics.  However, all too often the math learning experiences of students with learning differences are devoid of reasoning and problem solving. Instead they experience mathematics as an exercise in memorizing facts and reproducing recently demonstrated skills and procedures. With good intentions, we often break big ideas down into bite-sized pieces and reduce problem solving to enacting a suggested approach.  When students with learning differences spend the bulk of their time practicing computations, they come to see math as a collection of unrelated facts and skills to memorize and master.  When we break down math concepts and procedures into bite-sized pieces they lose their meaning and students struggle to make sense of and remember them. When we preemptively explain a word problem and suggest a solution path, we lower the demand and rob students of the opportunity to think and reason. Imagine if instead we placed the math thinking front and center.  We have found that when you place a premium on mathematical thinking, it serves to support student sense-making and math learning — especially for students with learning disabilities. It turns out that it’s impossible to focus on the thinking without bringing meaning into the mathematics.  Let us explain.

What is mathematical thinking?

We look to the Common Core math practice standards to define thinking and reasoning mathematically because they articulate the ways in which mathematicians work. Three math practices in particular describe how mathematicians think and reason: MP2 Reason abstractly and quantitatively,  MP7 Look for and make use of structure, and MP8 Look for and express regularity in repeated reasoning.  We affectionately refer to SMPs 2, 7, and 8 as “avenues of thinking” because they describe three distinct ways that productive math doers make sense of and make their way through prickly math problems, as well as make sense of and make use of important math concepts. Each of these avenues of thinking position students to interact with complex mathematical tasks in ways that often play to their learning strengths.  Let’s take a look at each avenue of thinking and how it might serve as a support for students with learning differences.  

How does mathematical thinking support students with learning disabilities?

The literature suggests that students with learning disabilities need math learning experiences that:

• Provide authentic, meaningful contexts, 
• Model learning strategies using multiple representations and multisensory techniques,
• Provide students with opportunities to use language to describe their understanding, and
• Provide multiple practice opportunities.

There is a great deal of overlap between these characteristics of effective learning experiences and learning experiences that prompt quantitative, structural, and repeated reasoning. For example, quantitative reasoning (MP2) draws heavily on real-world contexts, structural thinking (MP7) is awash in connecting multiple representations and strategies, and repeated reasoning (MP8) provides multiple practice opportunities as math doers look for regularity in calculating, counting and constructing while engaging in multi-modal experiences.  All three avenues of thinking place a premium on language to communicate reasoning.  Watch Amy’s NCTM Ignite video  and/or read chapter 2 in Routines for Reasoning for more about this overlap.

Further, each of the three avenues of thinking position students with learning differences to interact with mathematics in ways that often play to their learning strengths like visual processing, rather than weaknesses like working memory. Let us explain.

Quantitative Reasoning

Quantitative reasoners pay attention to quantities (the things that can be counted or measured) and the relationships between them in problem situations, real-world contexts and mathematical representations.  They ask themselves questions like, “What can I count or measure in this situation?”, “Is that number a value for a quantity or does it describe a relationship?”, and  “How can I represent this situation so that I can surface hidden quantities and see relationships?”.  Quantitative reasoning plays to the strengths of students who benefit from working within familiar contexts or benefit from drawing or using visual representations. Quantitative reasoning supports students who don’t know where to begin to solve a word problem or struggle with multi-step problems. Making sense of a problem by identifying important quantities and relationships orients and provides struggling students traction into complex problems.  

Structural Thinking

Structural thinkers pay attention to the big picture, they consider types of mathematical objects (e.g. numbers, shapes, graphs, equations, etc.) and leverage the rules for operating with them (e.g. properties of operations, geometric relationship, etc.)  Structural thinkers chunk complicated math objects into pieces they recognize, they change the form of numbers, expressions, irregular shapes, etc. to make them easier to work with, and they connect mathematical ideas and representations to make meaning. Structural thinking plays to the strength of students who see the “big picture” but often become befuddled with the details.  Because structural thinkers look for and leverage shortcuts, this type of thinking is a support for students who lose track of their work and/or calculations. With its emphasis on connections, structural thinking supports students who benefit from multiple representations, as well as students who struggle with conceptual processing.  Structural thinking helps students see that mathematics is interconnected and makes sense, rather than a collection of unrelated facts and procedures to memorize.

Repeated Reasoning 

Repeated reasoners pay attention to repetition in mathematical processes. Rather than looking for number patterns in outcomes, they look for repetition in the counting, calculating, and constructing processes that generate the outcomes. Because repeated reasoners record and track the process they engage in, repeated reasoning plays to the strengths of students who are organized and work in systematic ways. Looking for regularity in mathematical processes engages the senses –e.g. hearing the rhythm in counting or feeling the regularity in building with manipulatives or seeing the same calculations repeat on the page– and this can play to the strength of strong visual, auditory, and/or tactual-kinesthetic processors. Repeated reasoning is a support for students who struggle to abstract and generalize or who benefit from seeing how rules are developed.

To learn more about supporting students with learning disabilities to think and reason mathematically join us in our upcoming 3-day intensive remote course, Essential Strategies for Teaching Students with Learning Disabilities to Think Mathematically.